Prediction of the toxic effects of (agro) chemical mixtures on organisms using simple time-based models

The lethal effect of a chemical acting alone can be predicted using the simple hyperbolic model, which relies on the chemicals' median lethal time (LT50). However, this model cannot be used to predict mixture toxicity, considering that toxicity in natural ecosystems often results from exposure to mixtures rather than single chemicals. The lethal time addition method was developed to calculate the LT50 of a pesticide mixture from the LT50 of its components. It enables the hyperbolic model to estimate the lethal effects of a mix of pesticides at various exposure times. The hyperbolic model, complemented by the lethal-time addition model, predicted the percentage mortality of Clarias gariepinus and Oreochromis niloticus exposed to binary and quaternary mixtures of atrazine, mancozeb, chlorpyrifos, and lambda-cyhalothrin and estimated the 96 hr LC50 of the pesticide mixture.


Introduction
Pollutants exist as a mixture in the aquatic environment. The main goal of ecotoxicology is to predict the toxic effects of contaminants in the environment [ 1 , 2 ]. While animal models like fish are commonly used in the laboratory to predict toxicity by studying their biological responses to toxicants, there are increasing calls for alternative techniques like statistical or mathematical models that do not necessitate the use of animals [3] . Besides, it is also not feasible to perform toxicity tests for a mixture to assess joint toxicity due to many pollutants.
Previous authors have shown that a hyperbolic model which utilises the Michaelis-Menten mathematical expression could adequately describe the lethality of a pollutant over time [4] . The hyperbolic model only depends on two variables: the toxicant concentration and the exposure time. It is a straightforward model that permits the prediction of mortality for any combination of concentration, and time, whether they result from pulsed, post-acute, or continuous exposure to toxicants. The median lethal time (LT 50 ) is a vital component of the hyperbolic model, which must be determined for each toxicant concentration. It can be estimated using probit, logit, or Weibull models in a time-to-death bioassay. Toxic effects for concentrations other than those tested could also be evaluated using a complementary log-to-log model to calculate all LT 50 values for a toxicant [1] .
A limitation of the hyperbolic model is that it is only used to predict mortality for single chemicals. It cannot predict the toxicity of a mixture of pollutants. This limitation is significant because toxicity in natural ecosystems often results from exposure to mixtures of toxicants rather than a single toxicant [ 5 , 6 ]. Pesticides are one such class of pollutants that co-exists in the environment. The aquatic ecosystem is a sink for pesticides [7] which enter the aquatic ecosystems through direct application, spray drift, surface runoff from soil/pavement, depositions (both dry and rainy), and urban/industrial discharges [8] .
Can the hyperbolic model predict the toxic effects of mixtures of toxicants/pesticides? The answer probably lies in exploring ways of computing the LT 50 of the mix of toxicants/pesticides from the LT 50 of the individual toxicants/pesticides in the combination. This study presents a method for calculating the LT 50 of binary and quaternary toxicant mixtures from the LT 50 of the individual toxicant components that make up the mixture. The estimated LT 50 of the mix can then be used in the hyperbolic model to predict the mortality of organisms at any time of exposure. Alternatively, the derived mixture LT 50 can be used with other models since they represent the median effect values for specific concentrations.

Bioassay of single pesticides
The primary data is the median lethal times (LT 50 s) of single pesticides, atrazine, mancozeb, chlorpyrifos, and lambda-cyhalothrin, obtained from a 96-hour acute toxicity study. Some of this data was published in an earlier study [9] . The LT 50 data and the corresponding concentration were fitted to a regression model to get the regression coefficients a and b so that the LT 50 of concentrations beyond those tested experimentally could be obtained.
Standard procedures were followed to perform the toxicity tests [10] . The range-finding test was used to establish the concentrations for the actual toxicity test. Ten C. gariepinus and O. niloticus fingerlings were exposed in duplicates to different pesticide concentrations in 4-litre plastic aquaria containing one litre of the test solutions. Mortality count and time-to-death were recorded every hour, while the LC 50 and LT 50 were estimated at the end of 96 h.

Calculation of pesticide mixture LT 50 using the lethal time addition model
A range of binary and quaternary pesticide mixture concentrations were selected, considering the toxicity (concentration range from the acute toxicity test) of the major component in the mixture. The   concentration range included concentrations expected to cause below and above 50% mortality. The concentration of each element in the pesticide mixture was then calculated based on their ratio. The LT 50 of the individual pesticide concentration in each mixture was then estimated using the regression equation In (LT 50 ) = a + b. In ( C ) [1] , where a and b are the regression coefficients obtained from the fitted acute toxicity data of the single pesticides.
Once the LT 50 of the individual pesticides in the mixture was estimated, the lethal time model Eq. (1) was used to calculate the LT 50 of the pesticide mixtures.
Eq. (1) shows the complimentary lethal time addition model Ca and Cb are the concentration of each pesticide (a and b) in the mixture, and LT a and LT b are the lethal time of the concentrations of the pesticides a and b Prediction of pesticide mixture mortality using the hyperbolic model The calculated pesticide mixture LT 50 was subsequently fitted into the hyperbolic model ( Eq. (2) to predict the percentage mortality of the mixture at different exposure duration. However, only the 96 hr LC 50 of the mixtures were subsequently estimated.
Eq. (2) shows the hyperbolic model [4] used to predict the percentage mortality.

Calculation of pesticide mixture median lethal concentration using concentration addition model
The 96 hr LC 50 of the pesticide mixtures were also estimated using the concentration addition (CA) model [11] ( Eq. (3) ), A range of percentage mortality was selected (5,15,35, 60, and 90%), and the lethal concentration LCx1 and LCx2 of the single pesticides that would produce this percentage mortality were identified from the probit analysis of the single pesticides. They were then combined using the CA model to get the lethal concentration of the mixture.
LCx mix is the lethal concentration of the mixture, while LCx1 and LCx2 are the lethal concentrations of the single pesticides 1 and 2 that would produce the same percentage of mortality x. P1 and P2 are the proportion of each pesticide 1 and 2 in the mixture.

Bioassay of binary and quaternary pesticide mixture
Acute toxicity test of the pesticide mixtures was performed. Pesticide mixtures were prepared based on each fish species' equitoxic pesticide ratio (ratio of the 96-hr LC50 of single pesticides) [12] . The ratio of the pesticides in the mixture and the stock solution is outlined below;  Stock solutions were diluted to prepare the test solutions of different concentrations. The same procedure described for studying the acute toxicity of single pesticides was followed to study the acute toxicity of binary and quaternary mixtures. The 96-hr LC 50 of the mixtures was estimated at the end of 96 h. The fingerlings' mean length and weight were 2.91 ±0.07 cm and 0.22 ±0.01 g, respectively, for C. gariepinus , 3.69 ±0.11 cm, and 0.45 ±0.04 g, respectively for O. niloticus . Sel. Conc-selected concentration.

Data analyses
The LT 50 s and LC 50 s of single pesticides and LC 50 s of the pesticide mixtures from the bioassay were estimated using probit analysis. Probit analysis and linear regression were computed with SPSS version 25. For the linear regression, the dependent variable (y) was the LT 50, and the independent variable (x) was the external chemical concentration C. The model deviation ratio (MDR) was used as a quantitative measure for the compliance between observed mixture toxicity and the toxicity predicted by the hyperbolic model and the concentration addition model [13] . The valid MDR values range from 0.5 to 2.

MDR = P redicted LCx mix
Obser v ed LCx mix

Bioassay of single pesticides
The 96-hour LC 50 of each pesticide and LT 50 of the concentrations used for the single pesticides are presented in Table 1 . At 96 h, not up to 50% mortality had occurred for the concentrations marked with asterisks; thus, the LT 50 > 96 h was extrapolated from the probit analysis.  Table 2 shows the parameters for the regression equation In(LT 50 ) = a + b. In( C ) obtained by linear regression on the natural logarithm transformed data of both y (lethal time) and x (pesticide concentration) variables ( Table 1 ). A good fit to the model was obtained, with r 2 values above 0.75 in all cases.

Calculation of the LT 50 of binary and quaternary pesticide mixture
The calculated LT 50 of each selected concentration of the binary and quaternary pesticide mixture is presented in Tables 3 and 4 for C. gariepinus and O. niloticus, respectively. Tables 5 and 6 show the predicted percentage mortality for C. gariepinus and O. niloticus, respectively, for the selected concentration at different exposure times.

Prediction of pesticide mixture toxicity using concentration addition model
The calculated lethal concentration of the mixtures (LCmix) for C. gariepinus and O. niloticus, respectively, that would cause the selected percentage mortality, is presented in Tables 7 and 8 . Tables 9 and 10 show the test concentrations and the percentage mortality recorded during the pesticide mixture bioassay. Table 11 compares the pesticide mixtures' predicted and experimental 96 hr LC 50 . As indicated by the MDR, the hyperbolic model's predictive capability was slightly better than the concentration addition model. MDR equal to 1 indicates perfect compliance between the predicted and observed toxicity of the mixture. MDR greater than1 suggests that the mixture is more toxic than predicted (i.e., an underestimation of mixture toxicity by the models), and MDR less than 1 means that the mixture is less harmful than expected (i.e., an overestimation of mixture toxicity by the models).

Discussion
Toxicants acting singly or jointly are categorised as highly toxic if the LC 50 is between 0.1 and 1 mg/L, moderately toxic if it is between 1 and 10 mg/L, and slightly toxic if it is between 10 and 100 mg/L [14] . The observed LC 50, in agreement with the LC 50 derived from both models indicates that the equitoxic ratio of atrazine-mancozeb was slightly toxic to C. gariepinus but moderately toxic to O. niloticus . Atrazine-chlorpyrifos was moderately toxic, while chlorpyrifos-lambda cyhalothrin was highly toxic to both species though the LC 50 data suggests O niloticus was more sensitive than C. gariepinus to the mixtures. The hyperbolic model also correctly classifies mancozeb-chlorpyrifos and mancozeb-lambda as moderately toxic to both species, though the CA model classed them as slightly toxic. Both models incorrectly categorised the atrazine-lambda toxicity to C. gariepinus as moderately rather than slightly toxic. The hyperbolic model differed in classifying quaternary mixture toxic to C. gariepinus .
The predicted values by the hyperbolic model may deviate from the observed results if the pesticides in the mixture interact with each other. The term interaction includes all forms of joint action that vary from effect addition, i.e., a more significant effect (synergistic, potentiating) or a lesser effect (antagonistic). Interaction affects the median lethal time of the mixture (a key variable  for the hyperbolic model). A previous study showed that synergism decreases time-to-death and lethal concentration, while antagonism increases time-to-death and lethal concentration [9] . The interaction may be toxicokinetic (TK) or toxicodynamic (TD). TK interactions can occur during which a toxicant alters the effective concentration of another, while TD interactions arise when a toxicant influences the organism's response to another toxicant [15] . The predictive accuracy of the CA model may be affected by the different modes of action (MOA) of the pesticides [16] . The concentration addition model is better suited for mixtures whose components share a similar mode of action [17] . This criterion may explain why it performed better than the hyperbolic model in predicting the chlorpyrifos-lambda-cyhalothrin mix, given that both insecticides act on the central nervous system. Wang et al. [18] opined that deviation of the mixture toxicity predicted by the CA model would occur when the components have significantly different slopes.
The deviations within the statistical range of the MDR variation are acceptable and indicate that the observed values were within a factor of 2 of the predicted values [ 13 , 16 ]. MDRs outside this range may provide additional information about a mixture. Cedergreen [16] earlier reported that MDRs may identify mixtures and species groups involved in synergistic (MDR > 2), additive (0.5 ≤MDR ≤2), and antagonistic (MDR < 0.5) interaction. Synergism may also be present in mixes with MDRs slightly below two [16] . It follows that most pesticide mixtures in this study did not interact but produced an additive effect. The exceptions were antagonism in catfish exposed to atrazine-lambda-cyhalothrin and tilapia exposed to mancozeb-lambda-cyhalothrin, estimated by both models; same with atrazinelambda-cyhalothrin in Nile tilapia and chlorpyrifos-lambda-cyhalothrin in catfish, estimated by the hyperbolic and CA model, respectively. Furthermore, the MDR suggests a synergistic interaction for the quaternary mixture in both species by both models; same with atrazine-chlorpyrifos in Nile tilapia (CA model), mancozeb-chlorpyrifos in catfish (CA model) and Nile tilapia (HM model), and chlorpyrifoslambda-cyhalothrin in catfish (HM model). Some of the predictions by the MDR values do not agree with the results of an earlier study that used the relative toxic unit (RTU), synergistic ratio, and survival analysis to evaluate the interaction of the same pesticide mixture in O. niloticus [9] . Atrazine-mancozeb, atrazine-lambda-cyhalothrin, and mancozeb-chlorpyrifos which were synergistic in O. niloticus going by the (RTU), synergistic ratio, and survival analysis is not in agreement with the MDRs of both HM and CA model which indicates the toxicity of the mixtures were additive. However the MDR of both models agrees with the other procedure in predicting the mancozeb-lambda-cyhalothrin as antagonistic. This inconsistency suggests that the use of the MDR to estimate mixture interaction may be limited. Understanding the impacts of pesticide mixtures is essential for the ecological risk assessment of pesticides since the hazard of an individual chemical may be lower than that of a mixture of chemicals [19] . This study showed how the hyperbolic model could play an important role. Then again, the model may be helpful in the pulse exposure assessment of pesticide mixtures to predict the latent effects of pulse exposure to pesticide mixtures once the LT 50 of the mixture after pulse exposure is known. The main advantage of using the hyperbolic model is that it can estimate the toxic effects of pulse or continuous exposure to a mixture at any concentration level or time of exposure, whereas the concentration addition only uses LC 50 values at fixed times of exposure. Other models enable the estimation of the harmful effects of single toxicants in organisms with exposure time and concentration but may require other parameters that are not readily available. For instance, the life expectancy reduction model [20] requires the organism's internal LC 50 , LT 50 , and average normal life expectancy. Similarly, the lethality in the time model [21] requires the internal concentration in the organism, rate constant, and bio-concentration factor to predict the toxicity of single toxicants. Others, like the single or multiple pulses model [22] , require data on the mortality at exposure time, mortality rate constant, and recovery time to predict toxicity.

Conclusion
This study showed that the hyperbolic model could predict the percentage mortality and lethal endpoints (LC 50 ) of pesticides or any toxicant mixture in fish with the complimentary lethal time addition model. The model can be used during the risk assessment of pesticide mixture in aquatic ecosystems.

Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.